


The Case of the Missing Mathematician

by kadnarim



Category: Thrilling Adventures of Lovelace and Babbage
Language: English
Status: Completed
Published: 2015-12-19
Updated: 2015-12-19
Packaged: 2018-05-07 14:33:26
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 1,067
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/5459933
Author URL: https://archiveofourown.org/users/kadnarim/pseuds/kadnarim
Summary: <blockquote class="userstuff">
              <p>In which a mathematician makes a minor mistake and a merry mechanical misadventure is had.</p>
            </blockquote>





	The Case of the Missing Mathematician

**Author's Note:**

  * For [VerySleepy](https://archiveofourown.org/users/VerySleepy/gifts).



Babbage crashed through the door to the room containing the difference engine, brandishing papers. "Ada! Come out; I need to talk to you!"

A series of clanking sounds emerged from the machine, and presently Ada emerged. "Yes?" she asked, bemusedly. "What is it?"

"A most unusual thing has occurred! Hamilton has disappeared!"

Ada's face became rapidly more grave. "Disappeared? Has he been kidnapped?"

"No, no — he literally disappeared. He was working on one of his mathematical proofs in the library when the maid came up to bring him tea. He was in the middle of excitedly talking about some new result — the woman was much too distraught to remember the details — when, mid-sentence, he disappeared! I've brought some of the papers he was working on — we've got to figure out what happened!"

Ada nodded, gathering up some papers and tools and making room on the table. "Let's see them."

Babbage spread them out. "I believe that this" — he pointed to a line — "was the last thing he wrote before he vanished."

Ada peered at the page. On it was written:

_i 2_ = _j 2_ = _k 2_ = _ijk_ = –1[1]   


At first Ada was stymied. "I'm not sure — hmmm. Wait." She grabbed a pen and rummaged through the piles of paper on the desk to find one with relatively little already on it. Ignoring Babbage's look of dismay when she grabbed a page of one of his manuscripts, she started scribbling, occasionally referring back to Hamilton's notes and muttering to herself. After a few minutes she looked up, an expression of slightly manic delight on her face.

"Look, Babbage! I thought that Hamilton was being excessively poetical in his ideas of imaginary numbers on top of imaginary numbers, but he actually managed to do it!" She pushed the paper over.

Babbage looked. "Hmmm...so if we take a vector that...but we specify real coefficients...and the conjugates are nonreciprocal!...Yes, I see. But that doesn't explain where he's disappeared to!"

"Oh, but I think it does. Hamilton wanted to invent a second axis of imaginary numbers, extending the complex plane into three dimensions. Last time he was here he complained that he could add and subtract his inventions quite sensibly, but he couldn't convince them to multiply or divide properly...then before we could discuss the mathematics any further he got distracted and offered to read us his poetry, at which point he had to be politely shooed out.

"He was successful — but he had to invent _two_ new imaginary axes; he couldn't do it with one. Three mutually orthogonal imaginary axes, plus the real numbers...Hamilton discovered a fourth dimension! I believe in his excitement he must have entered it and gotten trapped."

Babbage's eyes lit up. "Aha! Then we must mount..."

**"...a rescue mission!"** Ada joined, the two grinning dangerously at one another.

"Hopefully he hasn't gone far. Perhaps if we return to the library —" Ada said, already heading off, "— we might be able to find some clues as to what direction he went in."

"Excellent!" Babbage answered, following.

Ada [peered] at the remaining scribblings on the desk, hoping to find more of Hamilton's thoughts on his mathematical system. Meanwhile, Babbage was looking around the room with a sudden sharp interest.

"Do you know, I suspect he might be even closer than we thought..." he murmured, and, louder, "Hamilton? Can you hear me?"

Ada was about to return to rummaging through stacks of paper when they heard a faint voice.

"...yes?...I...I'm not sure where I am...I seem to have gotten turned around somehow..."

"Not to worry! We'll get you out," replied Babbage brightly.

"Can you tell us anything else about what you did?" Ada asked.

"...in my excitement at...discovering the fourth dimension...I think I accidentally jumped into it...I had already taken several steps...before I quite realized what happened...I tried to retrace those steps but...only became more lost."

Ada and Babbage shared a glance. "I think I can fix this, if you tell me as much as you can remember about how you fell," Ada said, adding, "And...stay where you are for now."

After writing down Hamilton's reply, she and Babbage went back to the [room with the machine], promising to return to Hamilton soon.

"I understand why Hamilton was unable to return by retracing his steps," Babbage said once they were out of earshot, "Multiplication in his new mathematics is non-commutative; he must not have undone his accidental movements in the correct order. But how could he have derived the equations and failed to realize this? He must have been extraordinarily disoriented by his fall.[2]"

"Indeed. In any case, his extraneous movements have made it much harder to determine how to bring him back. Making this a job for...

**"The Difference Engine."**

Ada and Babbage cut up Ada's notes and calculations and equations, feeding them into the machine. Whirring and clicking, it eventually spat out its answer. Ada brought it back upstairs, calling out, "Hamilton? I can bring you back, if you follow my instructions exactly."

"...okay..."

And presently, a flustered but unharmed mathematician reappeared in the library. "Thank you!" Hamilton said.

Babbage smiled widely. "What an adventure!" He pulled a still-visibly flustered Hamilton downstairs, sitting him at a table.

Hamilton out of earshot, Babbage turned to Ada and winked. "Once again, the Difference Engine has saved the day."

[1] The equation above is indeed Hamilton's original moment of genius. While he didn't disappear, he nevertheless managed to incorporate a stroke of poetic drama into the affair. Hamilton was on a walk with his wife along the Royal Canal on October 16, 1843, when the idea of a four-dimensional complex space occurred to him. He carved the equation into the Brougham Bridge, where a plaque commemorating the event remains today.*

[2] "Non-commutative" is a fancy way of saying "depends on the order". Modern math and physics often have equations where the answer depends on the order you multiply things in, but at the time such systems were quite unusual. (For an example of non-commutative rotations, take a book and flip it up to face you, then give it a quarter turn clockwise. Next, start over but reverse the order.)

Still, my apologies go to Hamilton, whom I am sure would have realized the implications even while disoriented after being thrown into the fourth dimension.*

**Author's Note:**

> I've borrowed and run with a scene from the comics in which Hamilton extends the complex plane into a third dimension (which, as Padua notes, is an artistic liberty and is not mathematically valid). In this fic, the characters are in a three-dimensional world (well, Pocket Universe) and Hamilton's system extends into the fourth dimension.
> 
> Hamilton's numbers are historically accurate! (I can't say the same about anything else in this story.) They're more commonly known as quaternions and have various applications, from spacecraft maneuvering to video game mechanics.
> 
> If you care about the math: everything is teeeeechnically true, but I'm cheating by mixing two senses of the quaternions. What Ada says is true: three perpendicular imaginary axes, plus the one regular one, add up to four dimensions. However, the way that quaternions are normally used is to describe rotations, and to do that we need to constrain the values that they can take. It doesn't actually make sense to combine those senses, but I'm going to hide here in the end notes and cry "artistic liberty! I did it for the story!" until the pitchfork-carrying mathematicians go away.


End file.
